Available online at www.sciencedirect.com
European Journal of Operational Research 193 (2009) 567–580
www.elsevier.com/locate/ejor
O.R. Applications
Measurement of multiperiod aggregative efficiency
K. Sam Park *, Kwangtae Park
Korea University, Business School, Anam-5Ga, Seongbuk, Seoul 136-701, South Korea
Received 11 August 2005; accepted 12 November 2007
Available online 22 November 2007
Abstract
This article proposes a new method for measuring an aggregative efficiency of multiple period production systems. Every organization
or firm generates a time series of data that represent its performances in the resource utilization and output production over multiple
periods, and often desires an aggregated measure of efficiency for several periods of interest. Data envelopment analysis (DEA) has
become an accepted and well-known approach to evaluating efficiency performance in a wide range of cases. However, most of the
DEA studies have dealt primarily with ways to gauge the efficiency of production in only a single period so this efficiency reflects the
insufficient or partial performance of multiple period productions. The new method is developed through extensions of the concept
of Debreu–Farrell technical efficiency and is applied to evaluating the efficiency of cable TV service units with 3-year data.
2007 Elsevier B.V. All rights reserved.
Keywords: Efficiency valuations; DEA; Time series data
1. Introduction
Data envelopment analysis (DEA), a nonparametric approach, has brought in possibilities for use in evaluating the efficiency performances of many different kinds of entities engaged in many different activities in many different contexts (e.g.,
Charnes et al., 1994; Cooper et al., 2000). Although great flexibility and extendibility exist, most of the DEA studies have
dealt primary with cross sectional data and measured relative efficiencies in a single period, usually one year. Exceptions are
window analysis in DEA (Charnes et al., 1985) and, under the umbrella of nonparametric approaches in econometric studies, Malmquist-type indexes of productivity (e.g., Caves et al., 1982; Fare and Grosskopf, 1996). Looking beyond the difference between their model details, we recocgnize that their common goal is to account for the changing patterns of
efficiency performances over several periods of time. However, these approaches, while vital and practically useful, do
not take into account an aggregated measure of efficiency for multiple period production systems.
Another exception is dynamic DEA. Nemoto and Goto (1999) proposed a dynamic DEA model to measure the overall
efficiency of a multiperiod production system. This overall efficiency can be viewed as price or economic efficiency. They
assumed perfect foresight with respect to the input costs over multiple periods and, within a usual production possibility
set, determined an intertemporal efficient frontier in the way of minimizing the aggregated cost incurred by using inputs
over time. However, even in a particular period, the assumption of exact costs of individual inputs is unrealistic (Thompson
et al., 1990, 1995). Moreover, the true monetary value (or exact discount factor) of an input in the time horizon remains
unknown in practice. Sueyoshi and Sekitani (2005) developed a method of how to measure returns to scale in the framework of the dynamic DEA of Nemoto and Goto (1999).
*
Corresponding author. Tel.: +82 2 3290 2611; fax: +82 2 922 1380.
E-mail address: [email protected] (K.S. Park).
0377-2217/$ - see front matter 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.ejor.2007.11.028
568
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
Sengupta (1995, 1999) developed different types of dynamic DEA models in which various possible scenarios of aggregating input costs over time were considered. These models seek to determine the optimal levels of inputs (as the decision
variables) over time. The computed optimal inputs are then used to determine the overall efficiency as the ratio of the two
composite costs; one incurred by using the actual levels of inputs and the other evaluated at the optimal levels of inputs.
Thus, as is similar to the Nemoto and Goto (1999) approach, it assumes that the future prices of inputs are available in
determining the overall efficiency. However, Sengupta (1999) further extended his original ideas to incorporating the uncertainty of future input prices in the measurement of overall efficiency, while Nemoto and Goto (1999) assumed exact input
prices.
We present a different attempt to measure the aggregative efficiency in the context of time serial data. To distinguish it
from the previous work, we refer to the methodology as multiperiod data envelopment analysis (MDEA). This does not
require any information on price data or preferential weightings of inputs and outputs over multiple periods, and yields a
multiperiod aggregative efficiency (MAE) that corresponds conceptually to a technical (but not price or economic) efficiency of multiperiod production units. The development of MDEA is based on the concept of Debreu–Farrell’s technical
efficiency measurement. Following Debreu (1951), who provided the first measure of efficiency, Farrell (1957) proposed a
nonparametric way of estimating technical efficiency, among others, on the bases of empirical input and output data. He
suggested measuring the efficiency by means of comparing a target production unit with the unit on the efficient frontier.
We extend this concept of efficiency measurement to multiperiod production units in order to arrive at MAE.
The paper is organized as follows. In the next section, we provide a motivating example and then extend the concept of
the Farrell measurement to the context of time serial data for single input and multiple outputs. After this has been done we
then put our ideas in a general and rigorously established form to accomplish MDEA. This is followed by an illustrative
application to the 3-year data on cable TV service units. Finally, a summary and a sketch of further research opportunities
conclude this paper.
2. Developments
2.1. Preliminaries
Table 1 shows a simple example where four decision making units (DMUs) produce different amounts of two outputs
and consume the same unit amounts of single input in two periods t = 1, 2. Basically, using an ordinary DEA we can
obtain the efficiency ratings of each DMU for individual periods. Listed in the last two columns, the results show that
DMU1 is efficient for both periods, DMU2 is efficient for period t = 1 but inefficient for t = 2, and the other two DMUs
are inefficient for both periods. These efficiency ratings obtained for individual periods are basically needed for efficiency
valuations but reflect partial performances of multiperiod production units.
This brings into play an important question as to how we can achieve a multiperiod aggregative efficiency, shortly
referred to as MAE. To carry out a panel data analysis, we assume underlying production technologies in which all of
a period’s input is expected to go into producing the output for the same period. We do not consider a special production
system where current input amounts might be used to produce future period outputs. Specifically, the MAE is measured in
a manner that a DMU’s performance in a particular period is compared with the performance of all DMUs in the same
period. The same way is actually taken in DEA to obtain the efficiency ratings in the last two columns of Table 1, but each
of these ratings signifies a single period efficiency.
The desirable MAE measurement is expected to have the following basic and important features:
(a) It is straightforward to make a clear distinction between an always efficient DMU and a sometimes efficient DMU for
all periods. For example, in Table 1 DMU1 is efficient for both periods, while DMU2 is efficient for the first period
but inefficient for the second period. If only the two periods are of interest, then DMU1 is always efficient but DMU2
is sometimes efficient. Obviously, over the two periods, DMU1 has no inefficiency (in terms of technical efficiency)
while DMU2 has inefficiency as 25% (= 1.25–1) of its output levels for the second period. In the DEA literature
Table 1
A simple example of time serial data
DMUs
1
2
3
4
Period t = 1
Period t = 2
Efficiency ratings in each period
Output1
Output2
Input
Output1
Output2
Input
t=1
t=2
3
5
2
2
5
3
4
2
1
1
1
1
5
4
2
2
3
2
2
2
1
1
1
1
1
1
1.25
2.0
1
1.25
1.5
1.5
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
569
(e.g., Cooper et al., 2000) it is the rule that only the DMU without any inefficiency receives the full (100%) efficiency,
otherwise it does not. Thus, it is reasonable that the full efficiency in terms of MAE is only given to the always efficient DMU.
(b) If a DMU has inefficiencies in some individual periods, it is necessary to provide information as to when and how
much this DMU is inefficient. Exhibition of when it is inefficient leads to a sharp discrimination between a sometimes
efficient DMU (e.g., DMU2 in Table 1) and an always inefficient DMU (e.g., DMUs 3 and 4). Demonstration of how
much inefficiencies occur is a standard practice in DEA, which is usually described in the form of a radial measure
and slacks (Cooper et al., 2000). Thus, a similar demonstration for the possible inefficiency amounts over time needs
to be made in terms of MAE.
(c) It is important to give a transparent interpretation of the MAE measure we will develop here. Again, the efficiency
measure obtained from an ordinary DEA model consists of a radial measure and slacks. Their interpretations are
clear. For instance, the radial measure refers to the proportion of inefficiency present in all inputs or all outputs.
A positive slack for a particular input stands for a further inefficiency in that input. The MAE measure will also consist of a redial measure and slacks over multiple periods, so their interpretations need to be similar to (or consistent
with) those in DEA.
2.2. Single input and multiple output situations
Let there be L periods, t = 1, . . . , L. Assume that in each period single input is utilized to produce s different outputs,
r = 1, . . . , s. For each time period, we denote the utilized input by xt = 1 and the vector of the produced outputs by
y t ¼ ðy t1 ; . . . ; y ts ÞT . We also denote the production possible set, for each period, by Xt = {(xt, yt)|xt can produce yt}. To treat
an output-oriented efficiency valuation for all periods, we further define an output correspondence set,
Nðx1 ; . . . ; xL Þ ¼ fðy1 ; . . . ; yL Þjðxt ; yt Þ 2 Xt ; t ¼ 1; . . . ; Lg:
One of the most typical settings of N could be a convex set,
(
)
n
n
X
X
t
t
1
L
t
t t
N ¼ ðy ; . . . ; y Þj8t : y 6
yj kj ;
kj ¼ 1; 8t; j : kj P 0 ;
j¼1
j¼1
T
where the column vector ytj ¼ ðy t1j ; . . . ; y tsj Þ represents the actual outputs produced by DMUj in the tth period, j = 1, . . . , n.
The lambda variables employed serve the convex combination of the actual data of n DMUs for each period. Note that this
concrete setting of N can be altered in various ways, as needed (e.g., Bogetoft et al., 2000).
T
Recall the vectors of output variables yt ¼ ðy t1 ; . . . ; y ts Þ , t = 1, . . . , L. This representation is a result of period-oriented
arrangements of the individual variables y tr . Alternatively, their factor-oriented arrangements yield
yr ¼ ðy 1r ; . . . ; y Lr ÞT ; r ¼ 1; . . . ; s. Denote the collection of yr for all r by (y1, . . . , ys). Then, (y1, . . . , ys) 2 N holds if and only
if (y1, . . . , yL) 2 N holds. Similarly, the alternative representation of the DMUj’s output data, ytj , becomes
T
yrj ¼ ðy 1rj ; . . . ; y Lrj Þ .
For each output factor r, we define a vector of time weights reflecting the relative importance of different periods,
qr ¼ ðq1r ; . . . ; qLr Þ. We then make qryr to be a weighted sum of the time serial output variables. Further denote lr to be
the factor weight (or price per unit amount) of output r. We can then define a virtual output,
s
X
r¼1
lr qr yr ¼
s
X
v r yr ;
r¼1
where vr = lrqr is the row vector of the composite weights.
The concept of the Debreu–Farrell efficiency measurement can now be represented by
Ps
v y
Psr¼1 r r ;
r¼1 vr yro
where ðy1 ; . . . ; ys Þ 2 N is an optimal output location that will lie on the efficient frontier of N. Thus the efficiency score of
the outputs (y1o, . . . , yso) produced by DMUo becomes the ratio of the maximal production location to the DMUo production location. Farrell (1957) developed a linear algebraic approach to implement this concept (in the context of cross sectional data) which, in fact, was closely related to the way used in the envelopment (primal) side of DEA.1 However, in order
to achieve MAE, it is difficult to follow the algebraic approach because we have a more complicated situation in that we
need identify part (or facet) of the efficient frontier over multiple periods, simultaneously, on which the ðy1 ; . . . ; ys Þ point
1
See Charnes et al. (1978, Section 4) for the relationship between the Farrell approach and the envelopment DEA model.
570
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
lies. In addition, a direct extension of DEA to time serial data may fail to obtain the desired MAE (see sub Section 2.4
below). We thus develop a different approach to achieve MAE as follows.
Given the values of weights vr precisely for all r, the maximal output location corresponds theoretically to the optimal
solution of the following model:
Ps
r¼1 vr yr
P
:
ð1Þ
constant max
s
vr ;8r
ðy1 ;...;ys Þ2N
r¼1 vr yro
The outer term ‘constant’ means that all the values of weights are given a priori. We can then think of model (1) as a linear
program so we can readily obtain the desired maximum location, together with the efficiency score of DMUo, from solving
the linear programming problem. This simple approach would be appropriate, if exact values of weights are available, and
the resulting measure can rather be viewed as price efficiency than technical efficiency. However, as mentioned in the introduction, it may be hard to assess exact weights in the sense of actual market prices over time.2
For reasons like these, we do not require any information on these weights and assume only positive values, vr P e,
where e > 0 is a non-Archimedean infinitesimal. Meanwhile, one might instead use e = 0 restricting the weight variables
to be nonnegative which, however, ignores the possible slacks as sources of inefficiency. We employ e > 0 to follow the definition of the Pareto–Koopmans efficiency (e.g., Cooper et al., 1999 and the associated references therein).
Since the weights are unknown, vr P e, the portrayal of model (1) poses the need for another task in order to determine
how we treat these unknown variables. Based on the possible max and min optimization principles, we can consider either
max or min strategy as the outer objective of (1) for the unknown variables. If we take max strategy, model (1) becomes a
max–max structure. When taking min strategy, it becomes a min–max structure as follows:
Ps
vr y
ð2Þ
min max Psr¼1 r :
vr Pe;8r ðy1 ;...;ys Þ2N
v
r¼1 r yro
Generally, the measurement principle encountered in DEA can be stated as follows: The efficiency of DMUo under evaluation is obtained from an assessment in the light of the most favorable weight scenario. The most favorable set of weights
minimizes the gap between the weighted sum of the outputs of DMUo and that of the maximal outputs that will lie on the
efficient frontier of N. Therefore, the DEA principle naturally fits the min–max strategy used in (2), but not the max–max
strategy.
Without altering the optimal solution, model (2) can readily be modified to
(
)
X
s
s
X
v r yr vy ¼1 :
ð3Þ
min max
vr Pe;8r ðy1 ;...;ys Þ2N
r¼1 r ro
r¼1
Only the relative values of weights are relevant for the efficiency measurement under (2). Thus, as shown in (3), we can use
the same notation of weights even though the original weight values are normalized so that the virtual output of DMUo
becomes unity.
The following theorem is helpful in resolving problem (3):
Theorem 1. Let C and D be non-empty closed convex sets in finite dimensional real spaces, respectively, and let K be a
continuous finite concave–convex function on C D. If either C or D is bounded, one has
inf sup Kðc; dÞ ¼ sup inf Kðc; dÞ;
d2D c2C
c2C d2D
where c and d are the vectors of decision variables, and we say that K is a concave–convex function if K(c, d) is a concave
function of c 2 C for each d 2 D and a convex function of d 2 D for each c 2 C.
Proof. See Rockafellar (1970, Sections 36 and 37). h
Corollary 1. Model (3) is equivalent to
(
)
X
s
s
X
max min
v r yr v y ¼ 1; vr P e; 8r ;
r¼1 r ro
ðy1 ;...;ys Þ2N vr
r¼1
ð4Þ
where the order of the min–max objective in (3) is reversed while the desired optimal solution remains unchanged.
2
Farrell (1957) distinguished between technical efficiency and price efficiency, but he confined his studies mainly to technical efficiency. Regarding price
efficiency, he showed a conceptual way of measuring price efficiency through Diagram 1 in his paper, where exact market prices were actually assumed.
However, he also noted, for the most part, the formidable difficulties involved in assessing price efficiency, for example, because of the varying motives of
buyers and sellers.
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
571
Ls
Proof. This
C = N so that ðyr Þ ¼ ðyP
and D = {(vr) =
1 ; . . . ; ys Þ 2 C R
P can be proven using Theorem 1. DenoteLs
(v1, . . . , vs)| vryro = 1; vr P e, "r} so that ðvr Þ 2 D R . Define K[(yr), (vr)] = vryr. Then, the defined C and D are
non-empty closed convex sets in RLs . Moreover, the production possibility set C = N is obviously bounded. The defined
K[(yr), (vr)] is a linear function of (yr) 2 C for every (vr) 2 D and a linear function of (vr) 2 D for each (yr) 2 C. Thus, K
becomes a continuous finite concave–convex function on C D. Therefore, we have
s
s
X
X
inf sup
vr yr ¼ sup inf
v r yr :
ðvr Þ2D ðy Þ2C
r
r¼1
ðyr Þ2C ðvr Þ2D r¼1
Now the objectives inf and sup can be replaced by min and max, respectively, because the constraint sets C and D are
both closed. This completes the proof. h
Lemma 1. The optimal objective function value of model (4) is the same as that of the following model:
(
)
s
X
max max / þ e
1sr 8r : /yro þ sr ¼ yr ; sr P 0 ;
ðy1 ;...;ys Þ2N /;sr
r¼1
ð5Þ
where the scalar variable / is sign free, the nonnegative column vector sr 2 RL signifies the set of slacks for output r, and
the row vector 1 2 RL is a sum vector with all components equal to unity.
Proof. Note that the inner max problem in (5) is dual to the inner min problem in (4). It is assumed in model (4) that a
vector (vr) will be selected to minimize the resultant objective function once (yr) has been chosen. This implies that the inner
min problem in (4) becomes a linear program once (yr) is selected from N. Thus we can apply the dual theorem of linear
programming to that linear program which undoubtedly has a finite optimal solution. Analogous interpretations apply to
model (5). Finally, both (4) and (5) take the same outer objective as max to choose (yr) from the bounded N set. Therefore,
the optimal objective values of (4) and (5) are equal. h
Lemma 2. Without altering the optimal solution, the max–max problem in (5) can be reduced to the following regular max
problem:
(
)
s
X
/þe
1sr 8r : /yro þ sr ¼ yr ; sr P 0 :
max
ðy1 ;...;ys Þ2N
r¼1
/;sr
Proof. This is obvious.
h
Now incorporating the definition of production possibility set (y1, . . . , ys) 2 N into the model in Lemma 2, we can finalize
our elaborations to accomplish an operational model for obtaining the efficiency score of DMUo. This results in the following linear programming problem:
max
subject to
/þe
Xn
s X
L
X
r¼1
y t kt
j¼1 rj j
Xn
kt
j¼1 j
str
ð6:1Þ
t¼1
str ¼ /y tro ;
¼ 1;
r ¼ 1; . . . ; s;
t ¼ 1; . . . ; L
t ¼ 1; . . . ; L;
ð6:2Þ
ð6:3Þ
with all variables to be nonnegative except for /. Practically, the need for according a specific value to e > 0 is avoided in
terms of a two-phase operation analogous to the way the so-called big M is handled when a set of artificial variables is
treated in ordinary linear programming. Specifically, we first maximize the / variable subject to the constraints in (6.2)
and (6.3). Next the sum of the slack variables are maximized with the same constraints but the optimal /* value obtained
from the first phase is substituted in place of /. See Arnold et al. (1998) for an extended use of the non-Archimedean element and detailed discussions of two-phase procedures in DEA and linear programming.
Since / P 1 in model (6), it follows that the condition for full (100%) efficiency of any DMUo becomes /* = 1 and the
sum of the optimal slacks equals zero. We refer to model (6) as MDEA under single input and multiple output situations,
and the resulting efficiency as MAE.
2.3. Illustrations
To illustrate how to use model (6) for measuring MAE, we recall the time serial data in Table 1. For evaluating DMU1,
model (6) is specified by
572
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
Table 2
The resulting MAE from MDEA
Ratings /*
DMUs
1
2
3
4
1
1
1.25
1.5
Period t = 1
Period t = 2
Peers
st
1
st
2
Peers
st
1
st
2
Itself
Itself
DMU1
DMU2
0
0
0.5
2
0
0
0
0
Itself
DMU1
DMU1
DMU1
0
1
2.5
2
0
1
0.5
0
max
/ þ eðs11 þ s12 þ s21 þ s22 Þ
s:t:
3k11 þ 5k12 þ 2k13 þ 2k14 s11 ¼ 3/;
5k11 þ 3k12 þ 4k13 þ 2k14 s12 ¼ 5/;
5k21 þ 4k22 þ 2k23 þ 2k24 s21 ¼ 5/;
3k21 þ 2k22 þ 2k23 þ 2k24 s22 ¼ 3/;
k11 þ k12 þ k13 þ k14 ¼ 1;
k21 þ k22 þ k23 þ k24 ¼ 1
with all variables nonnegative except for /. Solving this problem we have a redial measure of efficiency /* and, for each
period, individual slacks, including referent DMUs (or peers) from the optimal lambda values. Similar operations can apply to the other DMUs. Table 2 summarizes the results for all DMUs.
As shown in Table 2 only DMU1 is fully efficient in terms of MAE. DMU2 is radially efficient (/* = 1, also called weakly
t
efficient in the DEA literature, e.g., Cooper et al., 1999) but not fully efficient because positive slacks ðst
1 ¼ s2 ¼ 1Þ occur in
period t = 2. This result implies that DMU2 is efficient for period t = 1 but inefficient for t = 2, which is consistent with the
result shown in Table 1. As expected, the other two DMUs appear to be inefficient in terms of MAE. It is worth noting
that, without employing an ordinary DEA, MDEA shows such desirable results for time serial data.
Therefore, it is confirmed that the developed MDEA model on the basis of the theoretical background has the necessary
and desirable features (a) through (c) mentioned previously. This is a result of the fact that MDEA takes the most optimistic redial measure of efficiency /* and identifies the remaining amounts of inefficiency in the form of positive slacks,
simultaneously. The optimistic radial measure means the minimal one of the maximum possible ratios in increasing the
respective sets of the individual period’s output data. For example, the /* of DMU2 is given by 1 = min {1, 1.25}. In fact,
in terms of (c), the maximum ratio by which the output data over all periods can simultaneously be increased is equivalent
to the optimistic radial measure.
An additional point to be noted is that, as in the ordinary DEA, multiple optimal solutions may occur on the reference
set of some DMUo in MDEA. For example, in Table 2, the peer of DMU4 was DMU2 for t = 1 so the slacks were iden1
tified by ðs1
1 ; s2 Þ ¼ ð5; 3Þ 1:5ð2; 2Þ ¼ ð2; 0Þ. See Table 1 for the data of DMUs. For the same t = 1, DMU1 can also be a
1
peer of DMU4 while maintaining the same optimal objective value that is a composition of / ¼ 1:5 and s1
1 þ s2 ¼ 2. In
1 1
this case the slacks are ðs1 ; s2 Þ ¼ ð3; 5Þ 1:5ð2; 2Þ ¼ ð0; 2Þ but the optimal objective value remains the same. Thus, Table 2
does not provide a complete set of peers for every DMU to be evaluated. However, our primary goal is to achieve an aggregative measure of efficiency along with identifying when and how much the evaluated DMU has inefficiencies, so we do not
pursue further the issue of multiple optimal solutions in DEA and MDEA.3
2.4. Remarks
We first note that taking a pessimistic radial measure of efficiency (e.g., 1.25 = max {1, 1.25} for DMU2 in Table 1) will
be unsuccessful to satisfy the desirable features of (b) and (c). This measure fails to clearly identify when DMU2 is inefficient and to sharply distinguish between a sometimes efficient DMU and an always inefficient DMU (i.e., violation of (b)).
3
In addition, the results in Tables 3, 5 and 6 of this paper may not be unique (because multiple optimal solutions may also be possible on peers and
slacks) except for the ‘‘ratings” in the first column of each of the tables. For details on how to treat the problem of multiple solutions in DEA, see the work
by Sueyoshi and Sekitani (2007a,b). They studied in-depth on the presence of multiple reference sets and multiple hyperplanes in several DEA models
including radial and nonradial additive models. They then proposed novel ways of identifying a complete set of peers and the type and magnitude of
returns to scale, which was done in relation to the concept of supporting hyperplanes. They did not deal with the time serial data envelopment model like
MDEA for those issues. Nevertheless, the treatments in their work could be incorporated into the framework of MDEA. For example, model (17) in
Sueyoshi and Sekitani (2007a) might be extended to the situation of time serial data so it could be used to determine a complete set of peers under MDEA.
Models (31) and (32) in the same article might also be extended so as to identify the type of returns to scale in MDEA.
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
573
This measure also fails to present a transparent interpretation as in the ordinary DEA because, within a usual production
possibility set, the first period’s output data for DMU2 cannot be increased by the ratio 1.25 (violation of (c)).
Second, one might consider a simple approach such as the average or weighted average of the individual-period efficiency ratings resulting from DEA. For example, the average efficiency of DMU2 becomes (1 + 1.25)/2 = 1.125. However,
as is similar to the pessimistic measure above, this average also violates (b) and (c). Moreover, the exact weights needed in
the weighted average scheme remain unknown in practice. While simple and intuitively plausible, it is thus difficult to say
that this approach has an appealing theoretical rationale.
In addition, we observe that if a DMU is weakly efficient in the ordinary DEA sense, then MDEA also renders this
DMU weakly efficient, because /* = 1 holds when the evaluated DMU is weakly efficient for at least one period. This consistency will be broken if we instead use the average or maximum value of the efficiency ratings obtained for individual
periods.
Third, one might consider a direct extension of DEA to the efficiency evaluation of time serial data. This idea can be
outlined as follows: For each input and output factor, first form a weighted sum of the serial data and, then, use these
weighted sums as the variables in a standard DEA. Suppose there are L periods of data for s output factors and a single
input factor, as above. If we use weighted sums of the variables, then the DEA would also have s outputs and one input.
Specifically, we may have
min
xðpxo Þ
subject to xðpxj Þ s
X
lr ðqr yrj Þ P 0;
j ¼ 1; . . . ; n;
r¼1
Xs
l ðq y Þ
r¼1 r r ro
¼ 1;
x; p; lr ; qr P e;
r ¼ 1; . . . ; s:
Here, the x and lr variables represent the input and output multipliers ordinarily employed in DEA. The variables in the
parenthesis signify the weighted sums of the serial data for every factor.
To continue with this idea, let u ¼ xp 2 RL andvr ¼ lr qr 2 RL . The immediately above model can then be reduced to
min
uxo
subject to uxj s
X
vr yrj P 0;
j ¼ 1; . . . ; n;
r¼1
Xs
r¼1
vr yro ¼ 1;
u; vr P e; r ¼ 1; . . . ; s:
T
We now consider dual to this model with the assumption of xj ¼ ð1; . . . ; 1Þ 2 RL for all j.
max
subject to
/þe
Xn
s X
L
X
r¼1
y t kj
j¼1 rj
Xn
j¼1
str
ð7:1Þ
t¼1
str ¼ /y tro ;
r ¼ 1; . . . ; s;
t ¼ 1; . . . ; L;
ð7:2Þ
kj ¼ 1
ð7:3Þ
with all variables nonnegative except for /.
The only difference between models (6) and (7) is that the current model employs kj instead of the ktj used in MDEA. By
using kj, the same output factors in different time periods are treated as if they were different output factors, so the total
number of output factors becomes s L. This entails a significantly less discrimination of the demonstrable inefficiencies
engaged in some periods. To make it more concrete, we refer to Table 3 in which we exhibit the resulting efficiency from
Table 3
The resulting efficiency from the DEA extension
DMUs
1
2
3
4
Ratings /*
1
1
1.25
1.5
Peers
Itself
Itself
DMU1
DMU1
Period t = 1
Period t = 2
st
1
st
2
st
1
st
2
0
0
0.5
0
0
0
0
2
0
0
2.5
2
0
0
0.5
0
574
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
using model (7) with the time serial data in Table 1. DMU2 appears to be fully efficient, while this DMU is not the case in
terms of MAE (see Table 2). Thus model (7) does not distinguish between, for all periods, an always efficient DMU and a
sometimes efficient DMU. This result also implies that model (7) may not provide useful information as to when and how
much a DMU is inefficient in individual periods. Consequently, such a direct extension of DEA to measuring a multiperiod
aggregative efficiency may lose the MDEA’s appealing features (a) and (b).
3. Generalization
3.1. Definitions
We now generalize the measurement concept and method examined in the single input and multiple output situations, as
in model (6), to multiple input and multiple output situations. Let there be L time periods, t = 1, . . . , L. Given a list of m
inputs xt 2 Rm and of s outputs yt 2 Rs , for each period, it is common practice in economic analysis to describe the activity
of a DMU by means of the production set Xt of attainable input and output points such as {(xt, yt)|xt can produce yt}. For
measuring MAE, we need to describe a unified production set X for all periods such that {[xt, yt]|xt can produce yt,
t = 1, . . . , L}. Here, [xt, yt] = [(x1, . . . , xL)T, (y1, . . . , yL)T] represents the collection (i.e., matrix) of the variable vectors xt
and yt for all t.
Suppose that we have a bundle of the input and output data pairs ðxtj ; ytj Þ; t ¼ 1; . . . ; L; empirically observed from
j = 1, . . . , n DMUs. Similarly let ½xtj ; ytj be the collection of the input and output data of DMUj for all t. One of the most
typical settings of the production possibility set is
(
)
n
n
n
X
X
X
t
t
t
t
t
t t
t
t t
xj kj ; y 6
yj kj ;
kj ¼ 1; 8t; j : kj P 0 :
ð8Þ
X ¼ ½x ; y j8t : x P
j¼1
j¼1
t
j¼1
t
Note that the representation of x and y is a result of period-oriented arrangements of the individual variables xti and y tr ,
respectively. Through factor-oriented arrangements, these individual variables can also be expressed as
L T
xi ¼ ðxt¼1
i ; . . . ; xi Þ ;
yr ¼
ðy 1r ; . . . ; y Lr ÞT ;
i ¼ 1; . . . ; m;
r ¼ 1; . . . ; s:
Similarly, the observed data xtj and ytj can alternatively be represented by
T
L
xij ¼ ðxt¼1
ij ; . . . ; xij Þ ;
yrj ¼
T
ðy 1rj ; . . . ; y Lrj Þ ;
i ¼ 1; . . . ; m;
r ¼ 1; . . . ; s;
j ¼ 1; . . . ; n;
j ¼ 1; . . . ; n:
With the collection of the rearranged variable vectors, [xi, yr] = [(x1, . . . , xm), (y1, . . . , ys)], and the rearranged data xij and
yrj, we can make the same production set as X defined in (8). This means that [xi, yr] 2 X holds if and only if [xt, yt] 2 X
holds.
For each input factor, we define a vector of time weights reflecting the relative importance of different periods,
pi ¼ ðp1i ; . . . ; pLi Þ. We then make pixi meaning a weighted sum of the time serial variables for input i. In the same way,
for each output factor we define qr ¼ ðq1r ; . . . ; qLr Þ and make qryr. Further denote xi and lr to be the factor weights of input
i and output r, respectively. We can then have a virtual input and a virtual output,
Xm
m
X
Xs
i¼1
s
X
xi pi xi ¼
i¼1
lqy
r¼1 r r r
¼
ui xi ;
v r yr ;
r¼1
where ui = xipi and vr = lrqr.
3.2. Output-oriented model
We now consider the following formulation:
Ps
Pm
ux
r¼1 vr yr Pmi¼1 i i :
min max Ps
½ui ;vr Pe ½xi ;yr 2X
r¼1 vr yro i¼1 ui xio
ð9Þ
If we have precise knowledge of the weights [ui, vr], then the outer min objective is unnecessary and formulation (9) becomes a linear program so we can readily obtain the optimal solution ½xi ; yr 2 X representing the maximal profit location
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
575
of inputs and outputs at the given weight information. Thus, as in the Debreu–Farrell price efficiency measurement, the
efficiency score of DMUo is given by the ratio of the maximal profit to the profit evaluated at [xio, yro], the input and output
data of DMUo to be evaluated.
However, we do not require any information about the weights and only assume their values positive, ui, vr P e, where
e > 0 is a non-Archimedean element. As illustrated in moving from model (1) and (2), we have thus taken the min strategy
for the weight variables in (9). Again, DEA assesses the DMUo efficiency in the light of the best scenario for weight values,
and the best weight scenario minimizes the inefficiency of DMUo (or the gap between the maximal profit and the DMUo
profit). Therefore, the measurement concept used in model (9) is consistent with that encountered in DEA.
While the optimal objective value remains unchanged, the fractional program in (9) can be modified to
(
)
X
s
m
s
m
X
X
X
v r yr ui x i vy ux ¼1 :
ð10Þ
min max
r¼1 r ro i¼1 i io
½ui ;vr Pe ½xi ;yr 2X
r¼1
i¼1
The original weight values are normalized so that the profit of DMUo becomes unity. For notational simplicity, we use the
same weight variables as those in (9).
Corollary 2. Model (10) is equivalent to
(
)
s
m
s
m
X
X
X
X
max min
v r yr ui xi vy u x ¼ 1; ui ; vr P e; 8i; r ;
r¼1 r ro i¼1 i io
½xi ;yr 2X ½ui ;vr r¼1
i¼1
ð11Þ
where the order of the min–max objective in (10) is reversed while the desired optimal solution remains the same.
Proof. This is immediate from Theorem 1 and Corollary 1. h
Lemma 3. The optimal objective function value of (11) is the same as that of the following model:
(
)
8r : /y þ s ¼ y ; s P 0;
s
m
X
X
r
ro
r r
max max / þ e
1sr þ e
1di ;
8i : /xio di ¼ xi ; di P 0
½xi ;yr 2X /;sr ;di
r¼1
i¼1
ð12Þ
where the scalar variable / is sign free and the nonnegative column vectors sr, di 2 RL represent the sets of slacks for output r
and input i, respectively. The row vector 1 2 RL in the objective is a sum vector with all components equal to unity.
Proof. This is immediate from Lemma 1.
h
Lemma 4. Without altering the optimal solution, the max–max problem in (12) can be reduced to the following regular max
problem:
s
m
X
X
1sr þ e
1di
ð13:1Þ
max
/þe
r¼1
subject to
i¼1
/yro þ sr ¼ yr ;
/xio di ¼ xi ;
r ¼ 1; . . . ; s;
i ¼ 1; . . . ; m;
ð13:2Þ
ð13:3Þ
½xi ; yr 2 X:
ð13:4Þ
h
Proof. This is obvious.
Now incorporating the production possibility set [xi, yr] 2 X defined in (8) into model (13), we can finalize our elaborations to accomplish an operational model for measuring MAE. This results in the following linear programming problem:
max
subject to
/þe
Xn
s X
L
X
r¼1
t¼1
str þ e
m X
L
X
i¼1
d ti
y t kt
j¼1 rj j
str ¼ /y tro ;
x t kt
j¼1 ij j
þ d ti ¼ xtio ; i ¼ 1; . . . ; m;
Xn
Xn
kt
j¼1 j
¼ 1;
ð14:1Þ
t¼1
r ¼ 1; . . . ; s;
t ¼ 1; . . . ; L
t ¼ 1; . . . ; L;
t ¼ 1; . . . ; L;
ð14:2Þ
ð14:3Þ
ð14:4Þ
with all variables nonnegative except for /. Note that because / is maximized in the objective, we can disregard its use for
the adjustment of inputs in the right of (14.3).
576
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
Therefore, we accomplish a linear programming model for measuring MAE in the context of time serial data for multiple inputs and outputs. We refer to model (14) as an output-oriented model of MDEA (see the next subsection for an inputoriented model). Using the concept of two-phase procedures in DEA, we can obtain the radial measure of efficiency /* and,
for each period, the optimal input and output slacks and the optimal lambda values to represent referent DMUs. Since /
P 1 in model (14), it follows that the condition for full efficiency of any DMUo becomes /* = 1 and the sum of the optimal
slacks equals zero. If a DMU is inefficient in terms of MAE, then we know from MDEA when and how much this DMU is
inefficient in individual periods.
To build model (14), we start with the production possibility set defined in (8) that is the convex hull of the actually
observed input and output data for all time periods. As a result, the driven model is under the assumption of variable
returns to scale for every time period. We can instead use another type of production set at the outset of the model building
such as the one under the constant returns to scale assumption. This removes the sum-to-unity conditions in (14.4) and
establishes another formulation that consists of (14.1) through (14.3). Other modifications or extensions are possible
according to which type of returns to scale is needed. On the returns to scale topic, see Cooper et al. (2000, Chapter 5)
and the corresponding references therein. See also Sueyoshi and Sekitani (2007a,b).
3.3. Input-oriented model
We have been dealing mainly with the so-called output-oriented model whose objective is to maximize outputs while
using no more than the observed amounts of any input. We can have another type of model that attempts to minimize
inputs while producing at least the given output levels, referred to as an input-oriented model. To accomplish an inputoriented model of MDEA, instead of formulation (9) we start with
Pm
Ps
v r yr
i¼1 ui xi Pr¼1
:
ð15Þ
max min Pm
s
½ui ;vr Pe ½xi ;yr 2X
i¼1 ui xio r¼1 vr yro
Unlike model (9), where the objective function is associated with a total profit (i.e., total revenue minus total cost) which is
maximized in the inner objective, model (15) employs a total loss (i.e., total cost minus total revenue) to be minimized in its
inner objective.
Given exact weights in (15), the outer max objective is not necessary and model (15) becomes a linear program so we can
readily obtain the optimal solution ½xi ; yr 2 X representing the minimal loss (or cost) location of inputs and outputs at the
given weight information. Thus, the efficiency score of DMUo is given by the ratio of the minimal cost to the cost evaluated
at [xio, yro], the input and output data of DMUo to be evaluated.
Since we assume the weights are unknown, ui, vrP e, we need to take either max or min strategy for the unknown weight
variables, as mentioned in moving from (1) and (2). We now take the max strategy for the weight variables in model (15).
This is consistent with the DEA measurement principle, because the newly taken max objective indeed minimizes the gap
between the minimal cost (as in the numerator) and the cost of DMUo (as in the denominator). This implies that the DMUo
efficiency is assessed in the light of the best scenario for weight values, which in turn maximizes the efficiency of DMUo.
The similar mathematical programming steps used to accomplish the output-oriented model in (14), can now be applied
to model (15). This results in the following input-oriented model:
min
subject to
/e
Xn
s X
L
X
r¼1
t¼1
str e
m X
L
X
i¼1
y t kt str ¼ y tro ;
j¼1 rj j
Xn
xt kt þ d ti ¼ /xtio ;
j¼1 ij j
Xn
kt
j¼1 j
¼ 1;
d ti
ð16:1Þ
t¼1
r ¼ 1; . . . ; s;
i ¼ 1; . . . ; m;
t ¼ 1; . . . ; L
t ¼ 1; . . . ; L;
t ¼ 1; . . . ; L;
ð16:2Þ
ð16:3Þ
ð16:4Þ
with all variables nonnegative except for /.
4. Illustrative application
4.1. Time serial data
In this section, we supply an illustrative application in which we show how to apply our developments. The example
involves evaluating efficiency performances of 20 cable TV (CATV) service operation units (SOs) in Korea. A total of
77 SOs existed at the end of 1999 and the sampled 20 are major SOs and most comparable. In other words, some SOs
excluded in this sample highly decrease comparability even in the case of major SOs. For instance, several SOs serve ‘‘home
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
577
shopping channels” so their ultimate goals are rather distribution of tangible goods. Some others are purely non-for-profit,
such as broadcasting servers for religion or education.
Important objectives in the CATV service operations include minimizing costs, maximizing revenues, and maximizing
subscribers. We thus use two inputs, operating cost (X1) and manpower (X2), and two outputs, revenue (Y1) and viewer
(Y2), in the efficiency evaluation. Operating cost consists of variable costs relevant to providing CATV services. It excludes
interest cost, depreciation of the equipment, and labor costs (because they are implicitly included in another input, manpower). Manpower represents the number of regular employees (or full-time equivalents). Revenue consists of the receipts
that are obtained by providing CATV services. The main sources of these receipts are from advertisement charges and subscription fees. Finally, viewer is the total number of CATV subscribers, which include charged subscribers as well as
uncharged subscribers. Shown in Table 4 are the time serial data for these inputs and outputs that contain 1999–2001 information on 20 SOs.
4.2. Evaluation results
We use model (14), an output-oriented MDEA model under the variable returns to scale (VRS) assumption, and summarize the resulting MAE for each of the 20 SOs in Table 5. This table provides the efficiency ratings (or radial measures of
efficiency) and, for each period, the slacks of inputs and outputs and the referent SOs. With the information, basically we
can classify every SO into three categories; fully efficient, weakly efficient, and inefficient. It is known from the MDEA
method that the fully efficient SOs are those that no inefficiency component has for each of the three years (i.e., always
efficient SOs for three years). If a SO is at least once inefficient in each year, then this SO is classified into either weakly
efficient or inefficient in terms of MAE. For each of the not fully efficient SOs, we can further identify when and how much
this SO is inefficient.
For example, SO1 appears to be weakly efficient in terms of MAE, because its radial measure equals unity but positive
slacks (12.7 for output 2, Y2, and 45.3 for input 1, X1) occur in 1999. These inefficiencies in 1999 are identified in comparison with referent SOs 11, 13, and 19. It is no wonder that SO13 is included as a referent DMU in 1999 because even
though not fully efficient for all periods, this SO is apparently efficient in 1999. We also know that, for each period, SO1 is
inefficient for 1999 but efficient for both 2000 and 2001, which in turn implies that the efficiency of SO1 is improved as time
elapsed. Analogous interpretations can apply to the other SOs.
In terms of MAE, eight SOs (3–6, 11, 14, 17, 19) appear to be fully efficient, three SOs (1, 2, 13) are weakly efficient, and
the other nine SOs are inefficient. Referring to the peer groups in the last three columns of Table 5, we find that SOs 17 and
19 are most frequently used to evaluate other SOs throughout the three periods. SO19 is used a total of 26 times to evaluate
Table 4
The 3-year data of CATV SOs
DMUs
Inputs
Outputs
Operating cost (100$)
SO1
SO2
SO3
SO4
SO5
SO6
SO7
SO8
SO9
SO10
SO11
SO12
SO13
SO14
SO15
SO16
SO17
SO18
SO19
SO20
Manpower (number)
Revenue (100$)
Viewer (100 numbers)
1999
2000
2001
1999
2000
2001
1999
2000
2001
1999
2000
2001
454
358
416
245
258
480
475
370
421
627
400
404
458
541
281
934
863
571
339
287
530
350
450
292
300
524
495
404
480
668
462
476
528
640
328
1018
1030
674
386
320
560
390
482
318
325
598
554
425
530
728
508
515
580
685
360
1102
1100
738
414
342
26
28
40
25
28
50
27
29
35
44
22
37
26
34
25
59
49
31
28
36
28
30
40
28
30
54
28
30
35
44
24
40
30
35
30
64
50
35
30
36
30
28
40
32
30
62
30
35
36
44
25
40
31
36
30
63
50
34
30
34
397
239
462
202
288
510
382
297
415
491
334
321
426
528
221
817
884
486
385
189
486
268
524
230
328
580
420
340
466
544
386
348
468
580
264
926
1020
568
462
215
580
312
620
286
405
612
504
399
529
624
454
400
520
696
305
1098
1104
660
554
254
186
105
244
116
149
311
198
161
201
264
170
168
210
320
127
361
424
258
196
90
360
160
445
208
270
545
346
269
325
442
320
300
368
526
252
615
828
482
365
180
396
176
480
226
292
565
368
275
335
469
352
325
389
570
264
665
900
520
400
188
578
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
Table 5
The resulting MAE under VRS assumption
DMUs
Ratings
SO1
SO2
1
1
SO3
SO4
SO5
SO6
SO7
SO8
Slacks
Referent DMUs
1999
2000
2001
1999
2000
2001
11, 13, 19
19
Itself
5, 19
Itself
Itself
Itself
Itself
Itself
Itself
11, 14, 17, 19
19
Itself
Itself
Itself
Itself
11, 17, 19
3, 19
1.120
1.302
1
1.400
1
Y2(12.7)
Y1(8)
0
0
0
3, 17, 19
6, 14, 17, 19
Itself
3, 6, 17, 19
Itself
3, 17, 19
3, 6, 17
Itself
3, 17, 19
11, 17, 19
3, 17, 19
3, 17
Itself
3, 17, 19
17, 19
SO14
SO15
SO16
1
1.066
1.005
Itself
4, 11, 19
17
Itself
5, 19
6, 17
Itself
5, 19
17
SO17
SO18
SO19
SO20
1
1.041
1
1.635
0
Y1(12)
Y1(62.5), Y2(61)
X1(71), X2(10)
0
X1(44.7)
0
Y1(13.7), Y2(18.7)
X2(8)
0
0
0
0
Y1(2.4), X1(125.4)
Y1(31.5), Y2(45.4)
X2(3.4)
Y1(58.5), Y2(119.4)
Y2(36.4)
0
Y1(85.2), Y2(45.5)
Y1(61.5), Y2(36)
X1(131.7)
0
Y1(138.5), Y2(53.1)
Y2(231.4)
X1(2), X2(13)
0
Y2(6.2), X1(22.4)
0
Y1(18.2), Y2(5.3)
X2(4)
Itself
Itself
Itself
Itself
13, 14, 19
6, 14, 19
SO9
SO10
SO11
SO12
SO13
0
Y1(137.9)
Y2(165.2)
0
0
0
0
0
Y1(7.6), Y2(5.5)
X1(18)
Y1(23.4), Y2(77.5)
Y1(1.8), Y2(14.3)
0
Y1(58.8), Y2(40.4)
Y2(3.2), X1(124.3)
0
0
1
1
1
1
1.098
1.336
Y2(12.7), X1(45.3)
Y1(146), Y2(91)
X1(19)
0
0
0
0
Y1(9.1), X1(27.7)
Y1(10.2)
Itself
13, 14, 17
Itself
5, 19
Itself
11, 17, 19
Itself
5, 6, 19
Itself
11, 17, 19
Itself
5, 19
0
Y1(90.2), Y2(32.3)
Y1(78.5), Y2(202.9)
X2(13.9)
0
Y1(30.6), X1(66.4)
0
X2(4.1)
other SOs, and SO17 is used 19 times. Thus these two SOs out of the eight fully efficient SOs can be recommended as benchmarking standards for the not fully efficient SOs.
Additionally, we carry out the efficiency evaluation using a different model, an output-oriented MDEA model under the
constant returns to scale (CRS) assumption, which is the model consisting of (14.1) through (14.3). Table 6 summarizes the
Table 6
The resulting MAE under CRS assumptiona
DMUs
SO1
SO2
SO3
SO4
SO5
SO6
SO7
SO8
SO9
SO10
SO11
SO12
SO13
SO14
SO15
SO16
SO17
SO18
SO19
SO20
a
Ratings
1.066
1.563
1
1.319
1.012
1
1.206
1.350
1.152
1.342
1.145
1.416
1.083
1
1.299
1.153
1
1.080
1
1.725
Referent DMUs
RTS statusb
Sum of lambdas
1999
2000
2001
1999
2000
2001
17, 19
14, 19
6, 19
6, 19
6, 19
Itself
14, 17
6, 14, 19
19
14, 17, 19
14, 17
6, 19
14, 17
Itself
6, 19
17, 19
Itself
14, 17
Itself
19
17, 19
19
Itself
19
19
Itself
17, 19
17, 19
17, 19
17, 19
17, 19
19
17, 19
17, 19
3, 6
17, 19
Itself
17, 19
Itself
3, 19
17, 19
17, 19
Itself
3
19
19
17, 19
19
17, 19
17, 19
17, 19
19
17, 19
17, 19
19
17, 19
Itself
17, 19
Itself
19
0.542
0.969
1.184
0.582
0.742
1
0.612
0.948
1.242
1.370
0.501
1.139
0.559
1
0.797
1.496
1
0.756
1
0.847
0.641
0.907
1
0.757
0.777
1
0.701
0.969
1.116
1.291
0.536
1.233
0.755
0.840
0.710
1.798
1
0.781
1
0.762
0.763
0.928
1
0.660
0.785
1.444
0.802
1.027
1.146
1.344
0.618
1.244
0.786
0.894
0.870
1.722
1
0.695
1
0.826
I-I-I
I-I-I
D-C-C
I-I-I
I-I-I
C-C-D
I-I-I
I-I-D
D-D-D
D-D-D
I-I-I
D-D-D
I-I-I
C-I-I
I-I-I
D-D-D
C-C-C
I-I-I
C-C-C
I-I-I
We omit listing the optimal slacks obtained from the computations.
This column represents the returns to scale status of each DMU for periods 1999–2001, where I Increasing, C Constant, and D Decreasing. For
example, I-I-I means triple Increasing sequentially for 1999, 2000, and 2001.
b
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
579
evaluation results. For almost all SOs, their radial measures of efficiency are drastically worsened in comparison with those
in Table 5. Now only two SOs (17 and 19), which are most frequently used to evaluate other SOs in Table 5, become fully
efficient in terms of MAE. In particular, we are interested in the changing pattern of returns to scale (RTS) statues in the
time horizon. It is natural that the two fully efficient SOs maintain constant RTS for all periods. Ten SOs (1, 2, 4, 5, 7, 11,
13, 15, 18, 20) show increasing RTS and their increasing RTS statuses remain unchanged as time passes. Four SOs (9, 10,
12, 16) appear to be decreasing RTS and their own statuses remain the same. However, the other four SOs (3, 6, 8, 14) show
varied RTS statuses. SO3 begins with decreasing RTS but changes to constant RTS in 2000, which is maintained in 2001.
SO6 changes from constant to decreasing, and SO8 jumps its production scale from increasing to decreasing. In contrast,
SO14 reduces its scale from constant to increasing.
5. Conclusions
Most of the previous DEA studies have dealt primarily with ways to evaluate the efficiency of production units in only a
single period, usually one year. In practice, however, almost all organizations generate panel data and may often desire an
aggregated measure of efficiency over several periods of time. Although DEA makes it possible to identify the efficiency
status and measure of a production unit for individual periods separately, it does not provide an aggregative efficiency status and measure over all periods of interest. Namely, we are supposed to obtain from DEA such information that a DMU
is efficient for a period while this DMU has some inefficiency for another period. The problem of rising importance would
then be how to combine these individual measures to represent an aggregated efficiency over the two periods. One might
take into account, for example, the average or weighted average of the individual measures. However, as noted in this
paper, such ways may yield an unsatisfactory result in that the interpretation of the average is not consistent with those
we have known from DEA and, moreover, the average approach fails to distinguish between sometimes efficient DMUs
and always inefficient DMUs over time. We have also shown that a direct extension of DEA to time serial data may lead
to an unsuccessful result, because it fails to make a distinction between always efficient DMUs and sometimes efficient
DMUs. Strictly, this approach renders a DMU fully efficient over all periods whenever this DMU is efficient for only
one period.
For these reasons, we have here developed a new approach to measuring an aggregative efficiency of multiple period
production units. This is based on the concept of Debreu–Farrell technical efficiency and the resulting linear programming
method is referred to as MDEA. MDEA enables to determine an aggregative efficiency measure in a single framework
upon which the efficiency classification can be based. The interpretation of the measure is similar to and consistent with
those in DEA. MDEA also provides when and how much production units are inefficient over time, which in fact makes
a clear distinction among always efficient, sometimes efficient, and always inefficient DMUs. In addition to the efficiency
classifications, MADE is capable of accounting for the change of efficiency over time such that a DMU’s efficiency is either
improved or worsened as time passes, which is shown through an application to cable TV service operation units. Furthermore, the structure of the MDEA model is quite similar to that of the standard DEA model so the application of MDEA
would be fairly easy if one is familiar with DEA. MDEA can be utilized to evaluate technical efficiency over time for many
different kinds of entities engaged in many different activities in many different contexts.
In practice, the management may often call for the evaluation result based on a broader analysis, for example, the recent
3-year data rather than just 1-year (or cross sectional) data. The reasons are: The evaluation using only one period data
might be unfair, because some units could be efficient or inefficient from time to time so according to the evaluation period
being fixed, one would be fortunate while another unfortunate. In sharp contrast, MDEA dealing with time serial data
provides an aggregative performance measure which classifies all these units equally fairly into a sometimes efficient group.
On the other hand, the consideration of multiperiod data in the performance evaluation may give rise to a more reliable
benchmark standard in that most frequently referred units over all periods of time would statistically be more reliable than
those for only one period. Furthermore, as shown through MDEA, the multiperiod data analysis is accountable for the
change of efficiency performances over time. This makes it possible to further prioritize the performance of some units
in the manner that the unit whose efficiency is on the rise is preferred to the other unit whose efficiency is falling, if the
two units are in the same efficiency class. Therefore, the developed MDEA method is supposed to meet the practical need
and expected to be of use for the management.
So far, we have dealt primarily with the assessment of technical efficiency with regard to when and how much production units are inefficient over multiple periods. Besides this, many other important issues may exist regarding the efficiency
evaluation in the context of time serial data, which include (i) treatment of multiple optimal solutions and (ii) identification
of targets.4 In Footnote 3 we set forth a possible way to address the problem in (i), but we leave its implementation as
future research. Regarding (ii), Portela et al. (2003) showed that one of the drawbacks of the traditional non-oriented
4
One of the referees pointed out that the two problems arise in MDEA.
580
K.S. Park, K. Park / European Journal of Operational Research 193 (2009) 567–580
DEA models (e.g., additive model) is that they aim at maximizing slacks, which contributes to finding targets and peers
that are not the closest to the unit being evaluated. The same problem might occur in MDEA, because its second phase
model corresponds to an additive model that maximizes sum of slacks. Efficiency improvement along the farthest path
may not be desirable for the unit being evaluated, so an extension or modification of the MDEA model needs to be made
in order to find the closest target, when the target setting is of particular interest. The idea of Portela et al. (2003) would be
helpful in doing this.
A final point to be noted is that we did not require any information about price data or preferential weightings of inputs
and outputs over time, so we measured a kind of technical efficiency of multiperiod production systems. This implies a clear
path for future researches on how we can measure a price or economic efficiency. The measurement of economic efficiency
calls for information on time preferences. However, this kind of weights relies heavily on subjective value judgments and
may often reflect evaluation policy of organization since one would never know the exact representation of time preferences. Thus, it is more realistic to assume the weight information is incomplete or imprecise, like the form of assurance
regions as in Thompson et al. (1990, 1995) and Charnes et al. (1990). In fact, the meanings between assurance region
and time preference information are different. Time preferences represent relative weights across different periods, while
assurance region conditions signify weights across different input or output factors in the same period. Nevertheless, we
can learn the treatment of such imprecise information from the assurance region approach. This also warrants further work
on the joint uses of assurance regions and time preferences in a unified manner to assess an extended measure of multiperiod overall efficiency.
Acknowledgements
We thank Professor Dyson, the Editor, and two anonymous referees for their helpful comments and suggestions, which
helped to improve this paper. We are also grateful for financial support from the IBRE of Korea University Business
School at Korea University.
References
Arnold, V., Bardhan, I., Cooper, W.W., Gallegos, A., 1998. Primal and dual optimality in computer codes using two-stage solution procedures in DEA.
In: Aranson, J., Zionts, S. (Eds.), Operations Research: Methods, Models and Applications. Quorum Books, Westport, CT, pp. 57–96.
Bogetoft, P., Tama, J.M., Tind, J., 2000. Convex input and output projections of nonconvex production possibility sets. Management Science 46 (6), 858–
869.
Caves, D., Christensen, L., Diewert, W.E., 1982. The economic theory of index numbers and the measurement of input, output, and productivity.
Econometrica 50 (6), 1393–1414.
Charnes, A., Cooper, W.W., Rhodes, E., 1978. Measuring the efficiency of decision making units. European Journal of Operational Research 2, 429–444.
Charnes, A., Clark, C.T., Cooper, W.W., Golany, B., 1985. A developmental study of data envelopment analysis in measuring the efficiency of
maintenance units in the U.S. air forces. Annals of Operations Research 2 (1), 95–112.
Charnes, A., Cooper, W.W., Huang, Z.M., Sun, D.B., 1990. Polyhedral cone-ratio DEA models with an illustrative application to large commercial
banks. Journal of Econometrics 46, 73–91.
Charnes, A., Cooper, W.W., Lewin, A.Y., Seiford, L.M., 1994. Data Envelopment Analysis: Theory, Methodology, and Application. Kluwer Academic
Publishers, Boston.
Cooper, W.W., Park, K.S., Pastor, J.T., 1999. RAM: A range adjusted measure of inefficiency for use with additive models, and relations to other models
and measures in DEA. Journal of Productivity Analysis 11, 5–42.
Cooper, W.W., Seiford, L.M., Tone, K., 2000. Data Envelopment Analysis: A Comprehensive Text with Models, Applications, References and DEASolver Software. Kluwer Academic Publishers, Boston.
Debreu, G., 1951. The coefficient of resource utilisation. Econometrica 19 (3), 273–292.
Fare, R., Grosskopf, S., 1996. Intertemporal Production Frontiers with Dynamic DEA. Kluwer Academic Publishers, Boston.
Farrell, M.J., 1957. The measurement of productive efficiency. Journal of the Royal Statistical Society (Series A) 120, 253–281.
Nemoto, J., Goto, M., 1999. Dynamic data envelopment analysis: Modeling intertemporal behavior of a firm in the presence of productive inefficiencies.
Economics Letters 64, 51–56.
Portela, M.C.A.S., Borges, P.C., Thanassoulis, E., 2003. Finding closest targets in non-oriented DEA models: The case of convex and non-convex
technologies. Journal of Productivity Analysis 19, 251–269.
Rockafellar, R.T., 1970. Convex Analysis. Princeton University Press.
Sengupta, J.K., 1995. Dynamics of Data Envelopment Analysis. In: Theory of Systems Efficiency. Kluwer Academic Publishers., Dordrecht.
Sengupta, J.K., 1999. A dynamic efficiency model using data envelopment analysis. International Journal of Production Economics 62, 209–218.
Sueyoshi, T., Sekitani, K., 2005. Returns to scale in dynamic DEA. European Journal of Operational Research 161, 536–544.
Sueyoshi, T., Sekitani, K., 2007a. The measurement of returns to scale under a simultaneous occurrence of multiple solutions in a reference set and a
supporting hyperplane. European Journal of Operational Research 181, 549–570.
Sueyoshi, T., Sekitani, K., 2007b. Measurement of returns to scale using a non-radial DEA model: A range-adjusted measure approach. European Journal
of Operational Research 176, 1918–1946.
Thompson, R.G., Langemeier, L.N., Lee, C.T., Lee, E., Thrall, R.M., 1990. The role of multiplier bounds in efficiency analysis with applications to
Kansas farming. Journal of Econometrics 46, 93–108.
Thompson, R.G., Dharmapala, P.S., Thrall, R.M., 1995. Linked-cone DEA profit ratios and technical efficiency with application to Illinois coal mines.
International Journal of Production Economics 39, 99–115.